Ch 10 - How Fast are you going? What time is it? 1 Basic...

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How Fast are you going? What time is it? 1 Basic Ideas of Interpolation ! Mathematical equivalent of “reading between the lines” ! Data - discrete samples of some function, f(x) ! Uses an interpolating function between points ! Data might… ! Exist as an experiment ! Analytic function that is difficult to evaluate 2
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Interpolation versus Curve Fitting (Regression) ! Interpolation ! Data assumed exact ! Interpolating function between points ! Curve passes EXACTLY through the data points ! Curve Fitting ! Uncertainty in data ! Uses a regression curve to fit data ! Curve passes “near” the points 3 Types of Interpolation ! Single Polynomial ! Divided Difference ! Lagrange Identical ! Multiple polynomials ! Hermite ! Lagrange 4
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Newton’s Divided-Difference is a Useful Form ! For “n” points, I can always fit an “n-1” degree polynomial ! Line b/w 2 points ! Parabola b/w 3 points ! 15 th order polynomial b/w 16 points ! Find coefficients of polynomial 21 01 2 1 () n n P xaa x a x a x ! ! "# # ## ± $% 12 1 1 2 1 ( ) ( ) nn Px b b x x b x x x x x x ! ! ! ! ! ±² Standard Form Newton’s Divided Difference 5 Linear Interpolation (Simplest Divided-Difference) 11 ( ) fx xx ! ! ! $ % 2 1 P xb b x x ! Imagine Similar Triangles: Newton-Form: x 1 x 2 x f(x 1 ) f(x 2 ) f(x) 1 2 1 2 1 1 ) ( ) ( ) ( ) ( x x x f x f x x x f x f ! ! " ! ! 6
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General Form- Newton’s Interpolating Polynomials $% 12 1 1 2 1 () ( ) ( ) nn Px b b x x b x x x x x x ! "# ! ## ! ! ! ±² &' 11 21 2 31 2 3 1 [, ] ,, ,,, , n bf x x x x x x x x x x !
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Ch 10 - How Fast are you going? What time is it? 1 Basic...

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